Table Of Content

NSF-ITP-94-97 hep-th/9411028 4 WHAT IS STRING THEORY? 9 9 1 v o N 3 1 Joseph Polchinski1 v 8 2 0 1 Institute for Theoretical Physics 1 University of California 4 9 Santa Barbara, CA 93106-4030 / h t - p e h : v i X ABSTRACT r a Lecturespresentedatthe1994 LesHouchesSummerSchool\FluctuatingGeome- triesinStatistical Mechanicsand FieldTheory." The (cid:12)rst part is an introduction to conformal (cid:12)eld theory and string perturbation theory. The second part deals with the search for a deeper answer to the question posed in the title. 1Electronic address: [email protected] Contents 1 Conformal Field Theory 5 1.1 The Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Mode Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 States and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Other CFT’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.7 Other Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.9 CFT on Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 String Theory 44 2.1 Why Strings? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 String Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4 The Weyl Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5 BRST Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.7 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.8 Trees and Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3 Vacua and Dualities 79 3.1 CFT’s and Vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Compacti(cid:12)cation on a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2 3.3 More on R-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4 N = 0 in N = 1 in :::? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5 S-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 String Field Theory or Not String Field Theory 98 4.1 String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Not String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3 High Energy and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5 Matrix Models 113 5.1 D = 2 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 The D = 1 Matrix Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Matrix Model String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 $ 5.4 General Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.5 Tree-Level Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.6 Spacetime Gravity in the D = 2 String . . . . . . . . . . . . . . . . . . . . . 130 5.7 Spacetime Gravity in the Matrix Model . . . . . . . . . . . . . . . . . . . . . 134 5.8 Strong Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3 While I was planning these lectures I happened to reread Ken Wilson’s account of his early work[1], and was struck by the parallel between string theory today and quantum (cid:12)eld theory thirty years ago. Then, as now, one had a good technical control over the perturbation theory but little else. Wilson saw himself as asking the question \What is quantum (cid:12)eld theory?" I found it enjoyable and inspiring to read about the various models he studied and approximations he tried (he refers to \clutching at straws") before he found the simple and powerful answer, that the theory is to be organized scale-by-scale rather than graph-by-graph. That understanding made it possible to answer both problems of principle, such as how quantum (cid:12)eld theory is to be de(cid:12)ned beyond perturbation theory, and practical problems, such as how to determine the ground states and phases of quantum (cid:12)eld theories. In string theory today we have these same kinds of problems, and I think there is good reason to expect that an equally powerful organizing principle remains to be found. There are many reasons, as I will touch upon later, to believe that string theory is the correct uni(cid:12)cation of gravity, quantum mechanics,and particle physics. It is implicit,then, that the theory actually exists, and ‘exists’ does not mean just perturbation theory. The nature of the organizing principle is at this point quite open, and may be very di(cid:11)erent from what we are used to in quantum (cid:12)eld theory. One can ask whether the situation today in string theory is really as favorable as it was for (cid:12)eld theory in the early 60’s. It is di(cid:14)cult to know. Then, of course, we had many more experiments to tell us how quantum (cid:12)eld theories actually behave. To o(cid:11)set that, we have today more experience and greater mathematical sophistication. As an optimist, I make an encouraging interpretation of the history, that many of the key advances in (cid:12)eld theory|Wilson’s renormalization group, the discovery of spontaneously broken gauge symmetry as the theory of the electroweak interaction, the discovery of general relativity itself|were carried out largely by study of simple model systems and limiting behaviors, and by considerations of internal consistency. These same tools are available in string theory today. My lectures divide into two parts|an introduction to string theory as we now under- stand it, and a look at attempts to go further. For the introduction, I obviously cannot in (cid:12)ve lectures cover the whole of superstring theory. Given time limitations, and given the broad range of interests among the students, I will try to focus on general principles. I will begin with conformal (cid:12)eld theory (2.5 lectures), which of course has condensed mat- ter applications as well as being the central tool in string theory. Section 2 (2.5 lectures) introduces string theory itself. Section 3 (1 lecture), on dualities and equivalences, covers 4 the steadily increasing evidence that what appear to be di(cid:11)erent string theories are in many cases di(cid:11)erent ground states of a single theory. Section 4 (1 lecture) addresses the question of whether ‘string (cid:12)eld theory’ is the organizing principle we seek. In section 5 (2 lectures) I discuss matrix models, exactly solvable string theories in low spacetime dimensions. I should emphasize that this is a survey of many subjects rather than a review of any single subject (for example R-duality, on which I spend half a lecture, was the subject of a recent review [2] with nearly 300 references). I made an e(cid:11)ort to choose references which will be useful to the student|a combination of reviews, some original references, and some interesting recent papers. 1 Conformal Field Theory Much of the material in this lecture, especially the (cid:12)rst part, is standard and can be found in many reviews. The 1988 Les Houches lectures by Ginsparg [3] and Cardy [4] focus on conformal (cid:12)eld theory, the latter with emphasis on applications in statistical mechanics. Introductions to string theory with emphasis on conformal (cid:12)eld theory can be found in refs. [5]-[9]. There are a number of recent books on string theory, though often with less emphasis on conformal techniques [10]-[14] as well as a book [15] and reprint collection [16] on conformal (cid:12)eld theory and statistical mechanics. Those who are in no great hurry will eventually (cid:12)nd an expanded version of these lectures in ref. [17]. Finally I should mention the seminal papers [18] and [19]. 1.1 The Operator Product Expansion The operator product expansion (OPE) plays a central role in this subject. I will introduce it using the example of a free scalar (cid:12)eld in two dimensions, X((cid:27)1;(cid:27)2). I will focus on two dimensions because this is the case that will be of interest for the string, and I will refer to these two dimensions as ‘space’ though later they will be the string world-sheet and space will be something else. The action is 1 S = d2(cid:27) (@ X)2 +(@ X)2 : (1.1.1) 1 2 8(cid:25) Z n o The normalization of the (cid:12)eld X (and so the action) is for later convenience. To be speci(cid:12)c I have taken two Euclidean dimensions, but almost everything, at least until we get to 5 nontrivial topologies, can be continued immediately to the Minkowski case1 (cid:27)2 i(cid:27)0. ! (cid:0) Expectation values are de(cid:12)ned by the functional integral < [X] > = [dX]e S [X]; (1.1.2) (cid:0) F F Z where [X] is any functional of X, such as a product of local operators.2 F It is very convenient to adopt complex coordinates z = (cid:27)1+i(cid:27)2; z(cid:22)= (cid:27)1 i(cid:27)2: (1.1.3) (cid:0) De(cid:12)ne also 1 1 @ = (@ i@ ); @ = (@ +i@ ): (1.1.4) z 1 2 z(cid:22) 1 2 2 (cid:0) 2 These have the properties @ z = 1, @ z(cid:22) = 0, and so on. Note also that d2z = 2d(cid:27)1d(cid:27)2 from z z the Jacobian, and that d2z(cid:14)2(z;z(cid:22)) = 1. I will further abbreviate @ to @ and @ to @(cid:22) when z z(cid:22) this will not be ambiguoRus. For a general vector, de(cid:12)ne as above 1 1 vz = v1+iv2; vz(cid:22) = v1 iv2; v = (v1 iv2); v = (v1+iv2): (1.1.5) z z(cid:22) (cid:0) 2 (cid:0) 2 For the indices 1, 2 the metric is the identity and we do not distinguish between upper and lower, while the complex indices are raised and lowered with3 1 g = g = ; g = g = 0; gzz(cid:22) = gz(cid:22)z = 2; gzz = gz(cid:22)z(cid:22) = 0 (1.1.6) zz(cid:22) z(cid:22)z zz z(cid:22)z(cid:22) 2 The action is then 1 S = d2z@X@(cid:22)X; (1.1.7) 4(cid:25) Z and the equation of motion is (cid:22) @@X(z;z(cid:22)) = 0: (1.1.8) The notation X(z;z(cid:22)) may seem redundant, since the value of z determines the value of z(cid:22), but it is useful to reserve the notation f(z) for (cid:12)elds whose equation of motion makes them 1Both the Euclidean and Minkowski cases should be familiar to the condensed matter audience. The former would be relevant to classical critical phenomena in two dimensions, and the latter to quantum critical phenomena in one dimension. 2Notice that this has not been normalized by dividingby <1>. 3Acommenton notation: beingcareful tokeep the Jacobian,one has d2z =2d(cid:27)1d(cid:27)2 and d2z detg = j j d(cid:27)1d(cid:27)2. However, in the literature one very frequently (cid:12)nds d2z used to mean d(cid:27)1d(cid:27)2. p 6 analytic in z. For example, it follows at once from the equation of motion (1.1.8) that @X is (cid:22) (cid:22) analytic and that @X is antianalytic (analytic in z(cid:22)), hence the notations @X(z) and @X(z(cid:22)). Notice that under the Minkowski continuation, an analytic (cid:12)eld becomes left-moving, a function only of (cid:27)0 +(cid:27)1, while an antianalytic (cid:12)eld becomes right-moving, a function only of (cid:27)0 (cid:27)1. (cid:0) Now, using the property of path integrals that the integral of a total derivative is zero, we have (cid:14) 0 = [dX] e SX(z ;z(cid:22)) (cid:0) 0 0 (cid:14)X(z;z(cid:22)) Z n o 1 = [dX]e S (cid:14)2(z z ;z(cid:22) z(cid:22))+ @ @ X(z;z(cid:22))X(z ;z(cid:22)) (cid:0) 0 0 z z(cid:22) 0 0 (cid:0) (cid:0) 2(cid:25) Z n o 1 = < (cid:14)2(z z ;z(cid:22) z(cid:22)) > + @ @ < X(z;z(cid:22))X(z ;z(cid:22)) > (1.1.9) 0 0 z z(cid:22) 0 0 (cid:0) (cid:0) 2(cid:25) That is, the equation of motion holds exceptat coincidentpoints. Now, the same calculation goes through if we have arbitrary additional insertions ‘:::’ in the path integral, as long as no other (cid:12)elds are at (z;z(cid:22)) or (z ;z(cid:22)): 0 0 1 @ @ < X(z;z(cid:22))X(z ;z(cid:22))::: > = < (cid:14)2(z z ;z(cid:22) z(cid:22))::: > : (1.1.10) z z(cid:22) 0 0 0 0 2(cid:25) (cid:0) (cid:0) (cid:0) A relation which holds in this sense will simply be written 1 @ @ X(z;z(cid:22))X(z ;z(cid:22)) = (cid:14)2(z z ;z(cid:22) z(cid:22)); (1.1.11) z z(cid:22) 0 0 0 0 2(cid:25) (cid:0) (cid:0) (cid:0) and will be called an operator equation. One can think of the additional (cid:12)elds ‘:::’ as preparing arbitrary initial and (cid:12)nal states, so if one cuts the path integral open to make an Hamiltoniandescription,an operator equation issimplyone whichholds for arbitrary matrix elements. Note also that because of the way the path integral is constructed from iterated time slices, any product of (cid:12)elds in the path integral goes over to a time-ordered product in the Hamiltonian form. In the Hamiltonian formalism, the delta-function in eq. (1.1.11) comes from the di(cid:11)erentiation of the time-ordering. Now we de(cid:12)ne a very useful combinatorial tool, normal ordering: :X(z;z(cid:22))X(z ;z(cid:22)): X(z;z(cid:22))X(z ;z(cid:22))+ln z z 2: (1.1.12) 0 0 0 0 0 (cid:17) j (cid:0) j 7 The logarithm satis(cid:12)es the equation of motion (1.1.11) with the opposite sign (the action was normalized such that this log would have coe(cid:14)cient 1), so that by construction @ @ :X(z;z(cid:22))X(z ;z(cid:22)): = 0: (1.1.13) z z(cid:22) 0 0 That is, the normal ordered product satis(cid:12)es the naive equation of motion. This implies that the normal ordered product is locally the sum of an analytic and antianalytic function (a standard result from complex analysis). Thus it can be Taylor expanded, and so from the de(cid:12)nition (1.1.12) we have (putting one operator at the origin for convenience) X(z;z(cid:22))X(0;0) = ln z 2 + :X2(0;0): + z :X@X(0;0): + z(cid:22):X@(cid:22)X(0;0): +::: : (1.1.14) (cid:0) j j This is an operator equation, in the same sense as the preceding equations. Eq. (1.1.14) is our (cid:12)rst example of an operator product expansion. For a general expec- tation value involving X(z;z(cid:22))X(0;0) and other (cid:12)elds, it gives the small-z behavior as a sum of terms, each of which is a known function of z times the expectation values of a single local operator. For a general (cid:12)eld theory, denote a complete set of local operators for a (cid:12)eld theory by . The OPE then takes the general form i A (z;z(cid:22)) (0;0) = ck (z;z(cid:22)) (0;0): (1.1.15) i j ij k A A A k X Later in section 1 I will give a simple derivation of the OPE (1.1.15), and of a rather broad generalization of it. OPE’s are frequently used in particle and condensed matter physics as asymptoticexpansions, the(cid:12)rstfewtermsgivingthedominantbehavioratsmallz. However, I will argue that, at least in conformally invariant theories, the OPE is actually a convergent series. The radius of convergence is given by the distance to the nearest other operator in the path integral. Because of this the coe(cid:14)cient functions ck (z;z(cid:22)), which as we will see ij must satisfy various further conditions, will enable us to reconstruct the entire (cid:12)eld theory. Exercise: The expectation value4 < X(z ;z(cid:22) )X(z ;z(cid:22) )X(z ;z(cid:22) )X(z ;z(cid:22) ) > is given by the 1 1 2 2 3 3 4 4 sum over all Wick contractions with the propagator ln z z 2. Compare the asymptotics i j (cid:0) j (cid:0) j as z z from the OPE (1.1.14) with the asymptotics of the exact expression. Verify that 1 2 ! the expansion in z z has the stated radius of convergence. 1 2 (cid:0) 4To be precise, expectation values of X(z;z(cid:22)) generally su(cid:11)er from an infrared divergence on the plane. This is a distraction which we ignore by some implicit long-distance regulator. In practice one is always interested in ‘good’operators such as derivatives or exponentials of X, which have well-de(cid:12)ned expectation values. 8 The various operators on the right-hand side of the OPE (1.1.14) involve products of (cid:12)elds at the same point. Usually in quantum (cid:12)eld theory such a product is divergent and must be appropriately cut o(cid:11) and renormalized, but here the normal ordering renders it well-de(cid:12)ned. Normal ordering is thus a convenient way to de(cid:12)ne composite operators in free (cid:12)eld theory. It is of little use in most interacting (cid:12)eld theories, because these have additional divergences from interaction vertices approaching the composite operator or one another. But many of the conformal (cid:12)eld theories that we will be interested in are free, and many others can be related to free (cid:12)eld theories, so it will be worthwhile to develop normal ordering somewhat further. For products of more than 2 (cid:12)elds the de(cid:12)nition (1.1.12) can be extended iteratively, :X(z;z(cid:22))X(z ;z(cid:22) ):::X(z ;z(cid:22) ): X(z;z(cid:22)) :X(z ;z(cid:22) ):::X(z ;z(cid:22) ): (1.1.16) 1 1 n n 1 1 n n (cid:17) + ln z z 2 :X(z ;z(cid:22) ):::X(z ;z(cid:22) ): + (n 1) permutations ; 1 2 2 n n j (cid:0) j (cid:0) n o contracting each pair (omitting the pair and subtracting ln z z 2). This has the same i (cid:0) j (cid:0) j properties as before: the equation of motion holds inside the normal ordering, and so the normal-ordered product is smooth. (Exercise: Show this. The simplest argument I have found is inductive, and uses the de(cid:12)nition twice to pull both X(z;z(cid:22)) and X(z ;z(cid:22) ) out of 1 1 the normal ordering.) The de(cid:12)nition (1.1.16) can be written more formally as (cid:14) :X(z;z(cid:22)) [X]: = X(z;z(cid:22)) : [X]:+ d2z ln z z 2 : [X]:; (1.1.17) 0 0 F F j (cid:0) j (cid:14)X(z ;z(cid:22)) F Z 0 0 for an arbitrary functional [X], the integral over the functional derivative producing all F contractions. Finally, the de(cid:12)nition of normal ordering can be written in a closed form by the same strategy, 1 (cid:14) (cid:14) : [X]: = exp d2zd2z ln z z 2 [X]: (1.1.18) 0 0 F 2 j (cid:0) j (cid:14)X(z;z(cid:22))(cid:14)X(z ;z(cid:22)) F (cid:26) Z 0 0 (cid:27) Theexponentialsumsoverallwaysof contractingzero,one,two,or morepairs. Theoperator product of two normal ordered operators can be represented compactly as (cid:14) (cid:14) : [X]: : [X]: = exp d2z d2z ln z z 2 F G : [X] [X]:; 0 00 0 00 F G (cid:0) j (cid:0) j (cid:14)X(z ;z(cid:22))(cid:14)X(z ;z(cid:22) ) F G (cid:26) Z 0 0 00 00 (cid:27) (1.1.19) 9 where(cid:14) and(cid:14) actonlyonthe(cid:12)eldsin and respectively. Theexpressions: [X]:: [X]: F G F G F G and : [X] [X]: di(cid:11)er by the contractions between one (cid:12)eld from and one (cid:12)eld from , F G F G which are then restored by the exponential. Now, for a local operator at z and a local 1 F G operator at z , we can expand in z z inside the normal ordering on the right to generate 2 1 2 (cid:0) the OPE. For example, one (cid:12)nds :eik1X(z;z(cid:22)): :eik2X(0;0): = z 2k1k2 :eik1X(z;z(cid:22))+ik2X(0;0): j j z 2k1k2 :ei(k1+k2)X(0;0): ; (1.1.20) (cid:24) j j since each contraction gives k k ln z 2 and the contractions exponentiate. Exponential op- 1 2 j j erators will be quite useful to us. Another example is ik @X(z;z(cid:22)) :eikX(0;0): :eikX(0;0): ; (1.1.21) (cid:24) (cid:0) z coming from a single contraction. 1.2 Ward Identities Theaction(1.1.7) has anumberofimportant symmetries,inparticularconformalinvariance. Let us (cid:12)rst derive the Ward identities for a general symmetry. Suppose we have (cid:12)elds (cid:30) ((cid:27)) (cid:11) with some action S[(cid:30)], and a symmetry (cid:30) ((cid:27)) = (cid:30) ((cid:27))+(cid:15)(cid:14)(cid:30) ((cid:27)): (1.2.1) 0(cid:11) (cid:11) (cid:11) That is, the product of the path integralmeasure and the weighte S is invariant. For a path (cid:0) integral with general insertion [(cid:30)], make the change of variables (1.2.1). The invariance of F the integral under change of variables, and the invariance of the measure times e S, give (cid:0) (cid:14) 0 = d2(cid:27) < (cid:14)(cid:30) ((cid:27)) [(cid:30)]> < (cid:14) [(cid:30)]> : (1.2.2) (cid:11) (cid:14)(cid:30) ((cid:27))F (cid:17) F Z (cid:11) (cid:11) X This simply states that the general expectation value is invariant under the symmetry. We can derive additional information from the symmetry: the existence of a conserved current (Noether’s theorem), and Ward identities for the expectation values of the current. Consider the following change of variables, (cid:30) ((cid:27)) = (cid:30) ((cid:27))+(cid:15)(cid:26)((cid:27))(cid:14)(cid:30) ((cid:27)): (1.2.3) 0(cid:11) (cid:11) (cid:11) 10

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What is string theory (hep-th 9411028) PDF

154 Pages·1994·0.695 MB·English
by  Polchinski J.
#Physics #Quantum Physics
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Preview What is string theory (hep-th 9411028)

NSF-ITP-94-97 hep-th/9411028 4 WHAT IS STRING THEORY? 9 9 1 v o N 3 1 Joseph Polchinski1 v 8 2 0 1 Institute for Theoretical Physics 1 University of California 4 9 Santa Barbara, CA 93106-4030 / h t - p e h : v i X ABSTRACT r a Lecturespresentedatthe1994 LesHouchesSummerSchool\FluctuatingGeome- triesinStatistical Mechanicsand FieldTheory." The (cid:12)rst part is an introduction to conformal (cid:12)eld theory and string perturbation theory. The second part deals with the search for a deeper answer to the question posed in the title. 1Electronic address: [email protected] Contents 1 Conformal Field Theory 5 1.1 The Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Conformal Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Mode Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 States and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Other CFT’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.7 Other Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8 Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.9 CFT on Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 String Theory 44 2.1 Why Strings? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 String Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4 The Weyl Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5 BRST Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.7 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.8 Trees and Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3 Vacua and Dualities 79 3.1 CFT’s and Vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Compacti(cid:12)cation on a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2 3.3 More on R-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4 N = 0 in N = 1 in :::? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5 S-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 String Field Theory or Not String Field Theory 98 4.1 String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2 Not String Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3 High Energy and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5 Matrix Models 113 5.1 D = 2 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 The D = 1 Matrix Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Matrix Model String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 $ 5.4 General Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.5 Tree-Level Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.6 Spacetime Gravity in the D = 2 String . . . . . . . . . . . . . . . . . . . . . 130 5.7 Spacetime Gravity in the Matrix Model . . . . . . . . . . . . . . . . . . . . . 134 5.8 Strong Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 3 While I was planning these lectures I happened to reread Ken Wilson’s account of his early work[1], and was struck by the parallel between string theory today and quantum (cid:12)eld theory thirty years ago. Then, as now, one had a good technical control over the perturbation theory but little else. Wilson saw himself as asking the question \What is quantum (cid:12)eld theory?" I found it enjoyable and inspiring to read about the various models he studied and approximations he tried (he refers to \clutching at straws") before he found the simple and powerful answer, that the theory is to be organized scale-by-scale rather than graph-by-graph. That understanding made it possible to answer both problems of principle, such as how quantum (cid:12)eld theory is to be de(cid:12)ned beyond perturbation theory, and practical problems, such as how to determine the ground states and phases of quantum (cid:12)eld theories. In string theory today we have these same kinds of problems, and I think there is good reason to expect that an equally powerful organizing principle remains to be found. There are many reasons, as I will touch upon later, to believe that string theory is the correct uni(cid:12)cation of gravity, quantum mechanics,and particle physics. It is implicit,then, that the theory actually exists, and ‘exists’ does not mean just perturbation theory. The nature of the organizing principle is at this point quite open, and may be very di(cid:11)erent from what we are used to in quantum (cid:12)eld theory. One can ask whether the situation today in string theory is really as favorable as it was for (cid:12)eld theory in the early 60’s. It is di(cid:14)cult to know. Then, of course, we had many more experiments to tell us how quantum (cid:12)eld theories actually behave. To o(cid:11)set that, we have today more experience and greater mathematical sophistication. As an optimist, I make an encouraging interpretation of the history, that many of the key advances in (cid:12)eld theory|Wilson’s renormalization group, the discovery of spontaneously broken gauge symmetry as the theory of the electroweak interaction, the discovery of general relativity itself|were carried out largely by study of simple model systems and limiting behaviors, and by considerations of internal consistency. These same tools are available in string theory today. My lectures divide into two parts|an introduction to string theory as we now under- stand it, and a look at attempts to go further. For the introduction, I obviously cannot in (cid:12)ve lectures cover the whole of superstring theory. Given time limitations, and given the broad range of interests among the students, I will try to focus on general principles. I will begin with conformal (cid:12)eld theory (2.5 lectures), which of course has condensed mat- ter applications as well as being the central tool in string theory. Section 2 (2.5 lectures) introduces string theory itself. Section 3 (1 lecture), on dualities and equivalences, covers 4 the steadily increasing evidence that what appear to be di(cid:11)erent string theories are in many cases di(cid:11)erent ground states of a single theory. Section 4 (1 lecture) addresses the question of whether ‘string (cid:12)eld theory’ is the organizing principle we seek. In section 5 (2 lectures) I discuss matrix models, exactly solvable string theories in low spacetime dimensions. I should emphasize that this is a survey of many subjects rather than a review of any single subject (for example R-duality, on which I spend half a lecture, was the subject of a recent review [2] with nearly 300 references). I made an e(cid:11)ort to choose references which will be useful to the student|a combination of reviews, some original references, and some interesting recent papers. 1 Conformal Field Theory Much of the material in this lecture, especially the (cid:12)rst part, is standard and can be found in many reviews. The 1988 Les Houches lectures by Ginsparg [3] and Cardy [4] focus on conformal (cid:12)eld theory, the latter with emphasis on applications in statistical mechanics. Introductions to string theory with emphasis on conformal (cid:12)eld theory can be found in refs. [5]-[9]. There are a number of recent books on string theory, though often with less emphasis on conformal techniques [10]-[14] as well as a book [15] and reprint collection [16] on conformal (cid:12)eld theory and statistical mechanics. Those who are in no great hurry will eventually (cid:12)nd an expanded version of these lectures in ref. [17]. Finally I should mention the seminal papers [18] and [19]. 1.1 The Operator Product Expansion The operator product expansion (OPE) plays a central role in this subject. I will introduce it using the example of a free scalar (cid:12)eld in two dimensions, X((cid:27)1;(cid:27)2). I will focus on two dimensions because this is the case that will be of interest for the string, and I will refer to these two dimensions as ‘space’ though later they will be the string world-sheet and space will be something else. The action is 1 S = d2(cid:27) (@ X)2 +(@ X)2 : (1.1.1) 1 2 8(cid:25) Z n o The normalization of the (cid:12)eld X (and so the action) is for later convenience. To be speci(cid:12)c I have taken two Euclidean dimensions, but almost everything, at least until we get to 5 nontrivial topologies, can be continued immediately to the Minkowski case1 (cid:27)2 i(cid:27)0. ! (cid:0) Expectation values are de(cid:12)ned by the functional integral < [X] > = [dX]e S [X]; (1.1.2) (cid:0) F F Z where [X] is any functional of X, such as a product of local operators.2 F It is very convenient to adopt complex coordinates z = (cid:27)1+i(cid:27)2; z(cid:22)= (cid:27)1 i(cid:27)2: (1.1.3) (cid:0) De(cid:12)ne also 1 1 @ = (@ i@ ); @ = (@ +i@ ): (1.1.4) z 1 2 z(cid:22) 1 2 2 (cid:0) 2 These have the properties @ z = 1, @ z(cid:22) = 0, and so on. Note also that d2z = 2d(cid:27)1d(cid:27)2 from z z the Jacobian, and that d2z(cid:14)2(z;z(cid:22)) = 1. I will further abbreviate @ to @ and @ to @(cid:22) when z z(cid:22) this will not be ambiguoRus. For a general vector, de(cid:12)ne as above 1 1 vz = v1+iv2; vz(cid:22) = v1 iv2; v = (v1 iv2); v = (v1+iv2): (1.1.5) z z(cid:22) (cid:0) 2 (cid:0) 2 For the indices 1, 2 the metric is the identity and we do not distinguish between upper and lower, while the complex indices are raised and lowered with3 1 g = g = ; g = g = 0; gzz(cid:22) = gz(cid:22)z = 2; gzz = gz(cid:22)z(cid:22) = 0 (1.1.6) zz(cid:22) z(cid:22)z zz z(cid:22)z(cid:22) 2 The action is then 1 S = d2z@X@(cid:22)X; (1.1.7) 4(cid:25) Z and the equation of motion is (cid:22) @@X(z;z(cid:22)) = 0: (1.1.8) The notation X(z;z(cid:22)) may seem redundant, since the value of z determines the value of z(cid:22), but it is useful to reserve the notation f(z) for (cid:12)elds whose equation of motion makes them 1Both the Euclidean and Minkowski cases should be familiar to the condensed matter audience. The former would be relevant to classical critical phenomena in two dimensions, and the latter to quantum critical phenomena in one dimension. 2Notice that this has not been normalized by dividingby <1>. 3Acommenton notation: beingcareful tokeep the Jacobian,one has d2z =2d(cid:27)1d(cid:27)2 and d2z detg = j j d(cid:27)1d(cid:27)2. However, in the literature one very frequently (cid:12)nds d2z used to mean d(cid:27)1d(cid:27)2. p 6 analytic in z. For example, it follows at once from the equation of motion (1.1.8) that @X is (cid:22) (cid:22) analytic and that @X is antianalytic (analytic in z(cid:22)), hence the notations @X(z) and @X(z(cid:22)). Notice that under the Minkowski continuation, an analytic (cid:12)eld becomes left-moving, a function only of (cid:27)0 +(cid:27)1, while an antianalytic (cid:12)eld becomes right-moving, a function only of (cid:27)0 (cid:27)1. (cid:0) Now, using the property of path integrals that the integral of a total derivative is zero, we have (cid:14) 0 = [dX] e SX(z ;z(cid:22)) (cid:0) 0 0 (cid:14)X(z;z(cid:22)) Z n o 1 = [dX]e S (cid:14)2(z z ;z(cid:22) z(cid:22))+ @ @ X(z;z(cid:22))X(z ;z(cid:22)) (cid:0) 0 0 z z(cid:22) 0 0 (cid:0) (cid:0) 2(cid:25) Z n o 1 = < (cid:14)2(z z ;z(cid:22) z(cid:22)) > + @ @ < X(z;z(cid:22))X(z ;z(cid:22)) > (1.1.9) 0 0 z z(cid:22) 0 0 (cid:0) (cid:0) 2(cid:25) That is, the equation of motion holds exceptat coincidentpoints. Now, the same calculation goes through if we have arbitrary additional insertions ‘:::’ in the path integral, as long as no other (cid:12)elds are at (z;z(cid:22)) or (z ;z(cid:22)): 0 0 1 @ @ < X(z;z(cid:22))X(z ;z(cid:22))::: > = < (cid:14)2(z z ;z(cid:22) z(cid:22))::: > : (1.1.10) z z(cid:22) 0 0 0 0 2(cid:25) (cid:0) (cid:0) (cid:0) A relation which holds in this sense will simply be written 1 @ @ X(z;z(cid:22))X(z ;z(cid:22)) = (cid:14)2(z z ;z(cid:22) z(cid:22)); (1.1.11) z z(cid:22) 0 0 0 0 2(cid:25) (cid:0) (cid:0) (cid:0) and will be called an operator equation. One can think of the additional (cid:12)elds ‘:::’ as preparing arbitrary initial and (cid:12)nal states, so if one cuts the path integral open to make an Hamiltoniandescription,an operator equation issimplyone whichholds for arbitrary matrix elements. Note also that because of the way the path integral is constructed from iterated time slices, any product of (cid:12)elds in the path integral goes over to a time-ordered product in the Hamiltonian form. In the Hamiltonian formalism, the delta-function in eq. (1.1.11) comes from the di(cid:11)erentiation of the time-ordering. Now we de(cid:12)ne a very useful combinatorial tool, normal ordering: :X(z;z(cid:22))X(z ;z(cid:22)): X(z;z(cid:22))X(z ;z(cid:22))+ln z z 2: (1.1.12) 0 0 0 0 0 (cid:17) j (cid:0) j 7 The logarithm satis(cid:12)es the equation of motion (1.1.11) with the opposite sign (the action was normalized such that this log would have coe(cid:14)cient 1), so that by construction @ @ :X(z;z(cid:22))X(z ;z(cid:22)): = 0: (1.1.13) z z(cid:22) 0 0 That is, the normal ordered product satis(cid:12)es the naive equation of motion. This implies that the normal ordered product is locally the sum of an analytic and antianalytic function (a standard result from complex analysis). Thus it can be Taylor expanded, and so from the de(cid:12)nition (1.1.12) we have (putting one operator at the origin for convenience) X(z;z(cid:22))X(0;0) = ln z 2 + :X2(0;0): + z :X@X(0;0): + z(cid:22):X@(cid:22)X(0;0): +::: : (1.1.14) (cid:0) j j This is an operator equation, in the same sense as the preceding equations. Eq. (1.1.14) is our (cid:12)rst example of an operator product expansion. For a general expec- tation value involving X(z;z(cid:22))X(0;0) and other (cid:12)elds, it gives the small-z behavior as a sum of terms, each of which is a known function of z times the expectation values of a single local operator. For a general (cid:12)eld theory, denote a complete set of local operators for a (cid:12)eld theory by . The OPE then takes the general form i A (z;z(cid:22)) (0;0) = ck (z;z(cid:22)) (0;0): (1.1.15) i j ij k A A A k X Later in section 1 I will give a simple derivation of the OPE (1.1.15), and of a rather broad generalization of it. OPE’s are frequently used in particle and condensed matter physics as asymptoticexpansions, the(cid:12)rstfewtermsgivingthedominantbehavioratsmallz. However, I will argue that, at least in conformally invariant theories, the OPE is actually a convergent series. The radius of convergence is given by the distance to the nearest other operator in the path integral. Because of this the coe(cid:14)cient functions ck (z;z(cid:22)), which as we will see ij must satisfy various further conditions, will enable us to reconstruct the entire (cid:12)eld theory. Exercise: The expectation value4 < X(z ;z(cid:22) )X(z ;z(cid:22) )X(z ;z(cid:22) )X(z ;z(cid:22) ) > is given by the 1 1 2 2 3 3 4 4 sum over all Wick contractions with the propagator ln z z 2. Compare the asymptotics i j (cid:0) j (cid:0) j as z z from the OPE (1.1.14) with the asymptotics of the exact expression. Verify that 1 2 ! the expansion in z z has the stated radius of convergence. 1 2 (cid:0) 4To be precise, expectation values of X(z;z(cid:22)) generally su(cid:11)er from an infrared divergence on the plane. This is a distraction which we ignore by some implicit long-distance regulator. In practice one is always interested in ‘good’operators such as derivatives or exponentials of X, which have well-de(cid:12)ned expectation values. 8 The various operators on the right-hand side of the OPE (1.1.14) involve products of (cid:12)elds at the same point. Usually in quantum (cid:12)eld theory such a product is divergent and must be appropriately cut o(cid:11) and renormalized, but here the normal ordering renders it well-de(cid:12)ned. Normal ordering is thus a convenient way to de(cid:12)ne composite operators in free (cid:12)eld theory. It is of little use in most interacting (cid:12)eld theories, because these have additional divergences from interaction vertices approaching the composite operator or one another. But many of the conformal (cid:12)eld theories that we will be interested in are free, and many others can be related to free (cid:12)eld theories, so it will be worthwhile to develop normal ordering somewhat further. For products of more than 2 (cid:12)elds the de(cid:12)nition (1.1.12) can be extended iteratively, :X(z;z(cid:22))X(z ;z(cid:22) ):::X(z ;z(cid:22) ): X(z;z(cid:22)) :X(z ;z(cid:22) ):::X(z ;z(cid:22) ): (1.1.16) 1 1 n n 1 1 n n (cid:17) + ln z z 2 :X(z ;z(cid:22) ):::X(z ;z(cid:22) ): + (n 1) permutations ; 1 2 2 n n j (cid:0) j (cid:0) n o contracting each pair (omitting the pair and subtracting ln z z 2). This has the same i (cid:0) j (cid:0) j properties as before: the equation of motion holds inside the normal ordering, and so the normal-ordered product is smooth. (Exercise: Show this. The simplest argument I have found is inductive, and uses the de(cid:12)nition twice to pull both X(z;z(cid:22)) and X(z ;z(cid:22) ) out of 1 1 the normal ordering.) The de(cid:12)nition (1.1.16) can be written more formally as (cid:14) :X(z;z(cid:22)) [X]: = X(z;z(cid:22)) : [X]:+ d2z ln z z 2 : [X]:; (1.1.17) 0 0 F F j (cid:0) j (cid:14)X(z ;z(cid:22)) F Z 0 0 for an arbitrary functional [X], the integral over the functional derivative producing all F contractions. Finally, the de(cid:12)nition of normal ordering can be written in a closed form by the same strategy, 1 (cid:14) (cid:14) : [X]: = exp d2zd2z ln z z 2 [X]: (1.1.18) 0 0 F 2 j (cid:0) j (cid:14)X(z;z(cid:22))(cid:14)X(z ;z(cid:22)) F (cid:26) Z 0 0 (cid:27) Theexponentialsumsoverallwaysof contractingzero,one,two,or morepairs. Theoperator product of two normal ordered operators can be represented compactly as (cid:14) (cid:14) : [X]: : [X]: = exp d2z d2z ln z z 2 F G : [X] [X]:; 0 00 0 00 F G (cid:0) j (cid:0) j (cid:14)X(z ;z(cid:22))(cid:14)X(z ;z(cid:22) ) F G (cid:26) Z 0 0 00 00 (cid:27) (1.1.19) 9 where(cid:14) and(cid:14) actonlyonthe(cid:12)eldsin and respectively. Theexpressions: [X]:: [X]: F G F G F G and : [X] [X]: di(cid:11)er by the contractions between one (cid:12)eld from and one (cid:12)eld from , F G F G which are then restored by the exponential. Now, for a local operator at z and a local 1 F G operator at z , we can expand in z z inside the normal ordering on the right to generate 2 1 2 (cid:0) the OPE. For example, one (cid:12)nds :eik1X(z;z(cid:22)): :eik2X(0;0): = z 2k1k2 :eik1X(z;z(cid:22))+ik2X(0;0): j j z 2k1k2 :ei(k1+k2)X(0;0): ; (1.1.20) (cid:24) j j since each contraction gives k k ln z 2 and the contractions exponentiate. Exponential op- 1 2 j j erators will be quite useful to us. Another example is ik @X(z;z(cid:22)) :eikX(0;0): :eikX(0;0): ; (1.1.21) (cid:24) (cid:0) z coming from a single contraction. 1.2 Ward Identities Theaction(1.1.7) has anumberofimportant symmetries,inparticularconformalinvariance. Let us (cid:12)rst derive the Ward identities for a general symmetry. Suppose we have (cid:12)elds (cid:30) ((cid:27)) (cid:11) with some action S[(cid:30)], and a symmetry (cid:30) ((cid:27)) = (cid:30) ((cid:27))+(cid:15)(cid:14)(cid:30) ((cid:27)): (1.2.1) 0(cid:11) (cid:11) (cid:11) That is, the product of the path integralmeasure and the weighte S is invariant. For a path (cid:0) integral with general insertion [(cid:30)], make the change of variables (1.2.1). The invariance of F the integral under change of variables, and the invariance of the measure times e S, give (cid:0) (cid:14) 0 = d2(cid:27) < (cid:14)(cid:30) ((cid:27)) [(cid:30)]> < (cid:14) [(cid:30)]> : (1.2.2) (cid:11) (cid:14)(cid:30) ((cid:27))F (cid:17) F Z (cid:11) (cid:11) X This simply states that the general expectation value is invariant under the symmetry. We can derive additional information from the symmetry: the existence of a conserved current (Noether’s theorem), and Ward identities for the expectation values of the current. Consider the following change of variables, (cid:30) ((cid:27)) = (cid:30) ((cid:27))+(cid:15)(cid:26)((cid:27))(cid:14)(cid:30) ((cid:27)): (1.2.3) 0(cid:11) (cid:11) (cid:11) 10

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